BinomialDist (Java Libraries for Stochastic Simulation)

java.lang.Object
- umontreal.iro.lecuyer.probdist.DiscreteDistributionInt
- - umontreal.iro.lecuyer.probdist.BinomialDist

All Implemented Interfaces:

Distribution
```
public class BinomialDist
extends DiscreteDistributionInt
```
Extends the class DiscreteDistributionInt for the binomial distribution with parameters n and p, where n is a positive integer and 0 <= p <= 1. Its mass function is given by

p(x) = nCr(n, x)p^x(1 - p)^n-x = n!/[x!(n - x)!] p^x(1 - p)^n-x for x = 0, 1, 2,…n,
and its distribution function is

F(x) = ∑_j=0^xnCr(n, j) p^j(1 - p)^n-j for x = 0, 1, 2,…n,
where nCr(n, x) is the number of possible combinations of x elements chosen among a set of n elements.

Field Summary

Fields
Modifier and Type Field and Description

static double MAXN
- Fields inherited from class umontreal.iro.lecuyer.probdist.DiscreteDistributionInt
  EPSILON

Fields
Modifier and Type	Field and Description
`static double`	`MAXN`

Constructor Summary

Constructors
Constructor and Description
`BinomialDist(int n, double p)` Creates an object that contains the binomial terms, for 0 <= x <= n, and the corresponding cumulative function.

Method Summary

Methods
Modifier and Type	Method and Description
`double`	`barF(int x)` Returns bar(F)(x), the complementary distribution function.
`double`	`cdf(int x)` Returns the distribution function F evaluated at `x` (see).
`static double`	`cdf(int n, double p, int x)` Computes F(x), the distribution function of a binomial random variable with parameters n and p, evaluated at x.
`static BinomialDist`	`getInstanceFromMLE(int[] x, int m)` Creates a new instance of a binomial distribution with both parameters n and p estimated using the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1.
`static BinomialDist`	`getInstanceFromMLE(int[] x, int m, int n)` Creates a new instance of a binomial distribution with given (fixed) parameter n, and with parameter p estimated by the maximum likelihood method based on the m observations x[i], i = 0, 1,…, m - 1.
`static double[]`	`getMaximumLikelihoodEstimate(int[] x, int m)` Deprecated.
`static double[]`	`getMaximumLikelihoodEstimate(int[] x, int m, int n)` Deprecated.
`double`	`getMean()` Returns the mean of the distribution function.
`static double`	`getMean(int n, double p)` Computes the mean E[X] = np of the binomial distribution with parameters n and p.
`static double[]`	`getMLE(int[] x, int m)` Estimates the parameters (n, p) of the binomial distribution using the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1.
`static double[]`	`getMLE(int[] x, int m, int n)` Estimates the parameter p of the binomial distribution with given (fixed) parameter n, by the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1.
`int`	`getN()` Returns the parameter n of this object.
`double`	`getP()` Returns the parameter p of this object.
`double[]`	`getParams()` Returns a table that contains the parameters (n, p) of the current distribution, in regular order: [n, p].
`double`	`getStandardDeviation()` Returns the standard deviation of the distribution function.
`static double`	`getStandardDeviation(int n, double p)` Computes the standard deviation of the Binomial distribution with parameters n and p.
`double`	`getVariance()` Returns the variance of the distribution function.
`static double`	`getVariance(int n, double p)` Computes the variance Var[X] = np(1 - p) of the binomial distribution with parameters n and p.
`static int`	`inverseF(int n, double p, double u)` Computes the inverse of the binomial distribution, x = F^-1(u), using a linear search starting at the mode if n is small.
`int`	`inverseFInt(double u)` Returns the inverse distribution function F^-1(u), where 0 <= u <= 1.
`double`	`prob(int x)` Returns p(x), the probability of x, which should be a real number in the interval [0, 1].
`static double`	`prob(int n, double p, double q, int x)` A generalization of the previous method.
`static double`	`prob(int n, double p, int x)` Computes and returns the binomial probability p(x) in eq..
`void`	`setParams(int n, double p)` Resets the parameters to these new values and recomputes everything as in the constructor.
`java.lang.String`	`toString()`

Methods inherited from class umontreal.iro.lecuyer.probdist.DiscreteDistributionInt
barF, cdf, getXinf, getXsup, inverseF

Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

- Field Detail
  - MAXN
```
public static double MAXN
```
- Constructor Detail
  - BinomialDist
```
public BinomialDist(int n,
            double p)
```
    Creates an object that contains the binomial terms, for 0 <= x <= n, and the corresponding cumulative function. These values are computed and stored in dynamic arrays, unless n exceeds MAXN.
- Method Detail
  - prob
```
public double prob(int x)
```
    Description copied from class: DiscreteDistributionInt
    
    Returns p(x), the probability of x, which should be a real number in the interval [0, 1].
    
    Specified by:
    
    prob in class DiscreteDistributionInt
    
    Parameters:
    x - value at which the mass function must be evaluated
    
    Returns:
    the mass function evaluated at x
  - cdf
```
public double cdf(int x)
```
    Description copied from class: DiscreteDistributionInt
    
    Returns the distribution function F evaluated at x (see).
    
    Specified by:
    
    cdf in class DiscreteDistributionInt
    
    Parameters:
    x - value at which the distribution function must be evaluated
    
    Returns:
    the distribution function evaluated at x
  - barF
```
public double barF(int x)
```
    Description copied from class: DiscreteDistributionInt
    
    Returns bar(F)(x), the complementary distribution function. See the WARNING above.
    
    Overrides:
    
    barF in class DiscreteDistributionInt
    
    Parameters:
    x - value at which the complementary distribution function must be evaluated
    
    Returns:
    the complementary distribution function evaluated at x
  - inverseFInt
```
public int inverseFInt(double u)
```
    Description copied from class: DiscreteDistributionInt
    
    Returns the inverse distribution function F^-1(u), where 0 <= u <= 1. The default implementation uses binary search.
    
    Overrides:
    
    inverseFInt in class DiscreteDistributionInt
    
    Parameters:
    u - value in the interval (0, 1) for which the inverse distribution function is evaluated
    
    Returns:
    the inverse distribution function evaluated at u
  - getMean
```
public double getMean()
```
    Description copied from interface: Distribution
    
    Returns the mean of the distribution function.
  - getVariance
```
public double getVariance()
```
    Description copied from interface: Distribution
    
    Returns the variance of the distribution function.
  - getStandardDeviation
```
public double getStandardDeviation()
```
    Description copied from interface: Distribution
    
    Returns the standard deviation of the distribution function.
  - prob
```
public static double prob(int n,
          double p,
          int x)
```
    Computes and returns the binomial probability p(x) in eq..
  - prob
```
public static double prob(int n,
          double p,
          double q,
          int x)
```
    A generalization of the previous method. Computes and returns the binomial term f (x) = (n!/x!(n-x)!)p^xq^n-x, where p and q are arbitrary real numbers (q is not necessarily equal to 1 - p). In the case where 0 <= p <= 1 and q = 1 - p, the returned value is a probability term for the binomial distribution.
  - cdf
```
public static double cdf(int n,
         double p,
         int x)
```
    Computes F(x), the distribution function of a binomial random variable with parameters n and p, evaluated at x.
  - inverseF
```
public static int inverseF(int n,
           double p,
           double u)
```
    Computes the inverse of the binomial distribution, x = F^-1(u), using a linear search starting at the mode if n is small. If n is larger than 10000, the linear search starts from 0 and the cdf static method is used to compute F(x) at different values of x, which much is less efficient.
  - getMLE
```
public static double[] getMLE(int[] x,
              int m)
```
    Estimates the parameters (n, p) of the binomial distribution using the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1. The estimates are returned in a two-element array, in regular order: [n, p].
    
    Parameters:
    x - the list of observations used to evaluate parameters
    m - the number of observations used to evaluate parameters
    
    Returns:
    returns the parameters [hat(n), hat(p)]
  - getMaximumLikelihoodEstimate
```
@Deprecated
public static double[] getMaximumLikelihoodEstimate(int[] x,
                                               int m)
```
    Deprecated.
    
    Same as getMLE.
  - getInstanceFromMLE
```
public static BinomialDist getInstanceFromMLE(int[] x,
                              int m)
```
    Creates a new instance of a binomial distribution with both parameters n and p estimated using the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1.
    
    Parameters:
    x - the list of observations to use to estimate the parameters
    m - the number of observations to use to estimate the parameters
  - getMLE
```
public static double[] getMLE(int[] x,
              int m,
              int n)
```
    Estimates the parameter p of the binomial distribution with given (fixed) parameter n, by the maximum likelihood method, from the m observations x[i], i = 0, 1,…, m - 1. Returns the estimator in an array with a single element.
    
    Parameters:
    x - the list of observations used to evaluate parameters
    m - the number of observations used to evaluate parameters
    n - the number of success
    
    Returns:
    returns the parameter [hat(p)]
  - getMaximumLikelihoodEstimate
```
@Deprecated
public static double[] getMaximumLikelihoodEstimate(int[] x,
                                               int m,
                                               int n)
```
    Deprecated.
    
    Same as getMLE.
  - getInstanceFromMLE
```
public static BinomialDist getInstanceFromMLE(int[] x,
                              int m,
                              int n)
```
    Creates a new instance of a binomial distribution with given (fixed) parameter n, and with parameter p estimated by the maximum likelihood method based on the m observations x[i], i = 0, 1,…, m - 1.
    
    Parameters:
    x - the list of observations to use to evaluate parameters
    m - the number of observations to use to evaluate parameters
    n - the parameter n of the binomial
  - getMean
```
public static double getMean(int n,
             double p)
```
    Computes the mean E[X] = np of the binomial distribution with parameters n and p.
    
    Returns:
    the mean of the Binomial distribution E[X] = np
  - getVariance
```
public static double getVariance(int n,
                 double p)
```
    Computes the variance Var[X] = np(1 - p) of the binomial distribution with parameters n and p.
    
    Returns:
    the variance of the binomial distribution Var[X] = np(1 - p)
  - getStandardDeviation
```
public static double getStandardDeviation(int n,
                          double p)
```
    Computes the standard deviation of the Binomial distribution with parameters n and p.
    
    Returns:
    the standard deviation of the binomial distribution
  - getN
```
public int getN()
```
    Returns the parameter n of this object.
  - getP
```
public double getP()
```
    Returns the parameter p of this object.
  - getParams
```
public double[] getParams()
```
    Returns a table that contains the parameters (n, p) of the current distribution, in regular order: [n, p].
  - setParams
```
public void setParams(int n,
             double p)
```
    Resets the parameters to these new values and recomputes everything as in the constructor. From the performance viewpoint, it is essentially the same as constructing a new BinomialDist object.
  - toString
```
public java.lang.String toString()
```
    Overrides:
    
    toString in class java.lang.Object

Class BinomialDist

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.DiscreteDistributionInt

Constructor Summary

Method Summary

Methods inherited from class umontreal.iro.lecuyer.probdist.DiscreteDistributionInt

Methods inherited from class java.lang.Object

Field Detail

MAXN

Constructor Detail

BinomialDist

Method Detail

prob

cdf

barF

inverseFInt

getMean

getVariance

getStandardDeviation

prob

prob

cdf

inverseF

getMLE

getMaximumLikelihoodEstimate

getInstanceFromMLE

getMLE

getMaximumLikelihoodEstimate

getInstanceFromMLE

getMean

getVariance

getStandardDeviation

getN

getP

getParams

setParams

toString